Cyclical formula? (noob warning)

[romanawgarlic]

Hello all! I am just a guy whos taking a basic statistics course and came across a phenomena that interested me. Probably very basic, but i can't figure it out by myself.

Was messing around on my TI-84 the other day, and ended up just pressing "solve" on a binomial probability function where the probability of the events were the answers of the previously performed function. I don't know exactly how to explain it, so i'll just write it out:

(10 nCr 1)(ans^9)((1-ans)^1)

weirdly, i kept getting numbers around the same values, so i put the equation into excell and ran it a thousand times with a random starting number (0<x<1). strangely, no matter what the first number was, the answers always tended to the same four numbers, in the same order:

0,385203336 0,048336317 0,30947461 0,110469374

What gives? Nothing about this in my textbook, and i don't know where to start searching on the internet. An explanation or link to related theory or anything that will alleviate my puzzlement would be much appreciated.

Thanks y'all!

 

[Office_Shredder]

Are you saying that you evaluated the function
$$10 x^9(1-x)$$
and kept getting the same numbers as you varied x?

 

[romanawgarlic]

Are you saying that you evaluated the function
$$10 x^9(1-x)$$
and kept getting the same numbers as you varied x?

sorry, got the equation mixed up! (feels like it shouldn't matter, but apparently it does when i plug it into excell

1. started by plugging a random number 0<x<1 into $$10 x(1-x)^9$$
2. took the answer i got from that and put it through the same function $$10 x(1-x)^9$$
3. took the answer from step 2 and put it through the equation again, and again, about a thousand times.


result: i ended up getting these numbers as answers time and time again in that order:
0,385203336 0,048336317 0,30947461 0,110469374

sorry about not being clear, I'm not great with mathematical terminology.

if you want to try it for yourself, you can put this in cell A3 in an excell sheet:

=RAND()

and this in cell A4

=10*POWER(A3,1)*POWER(1-A3,9)

then repeat the formula in the cell of A4 as many times as you need (about 100 times should do it)

thanks

 

[willem2]

With a first power instead of a ninth power, this iteration is known as the logistic map

http://mathworld.wolfram.com/LogisticMap.html

It seems to have a lot of the same properties

try to graph y=f(f(f(f(x)))) with f(x) = 10 x(1-x)^9. The graph will intersect in 4 places with the graph of y = x, which are the points of the cycle you've found